More details regarding the estimation protocol, as well as a rigorous proof, are provided in Appendix REF .
Theorem REF  improves upon existing shadow tomography protocols [1]}, [2]}, [3]}, [4]} for the special case of predicting Pauli observables.
Because each access to \(\operatorname{{\mathcal {E}}}_\rho \)  allows us to obtain one copy of \(\rho \) , we only need \(N_{\mathrm {Q}} = \mathcal {O}(n)\)  to predict expectation values of all \(4^n\)  Pauli observables up to a constant error.
